outputs/claim1_graph_dimension.json

{
  "gamma": 0.1,
  "seed": 20260725,
  "checks": [
    {
      "name": "full boolean cube on 2 points",
      "computed_d_gamma": 2,
      "expected_d_gamma": 2,
      "match": true,
      "note": ""
    },
    {
      "name": "full boolean cube on 3 points",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": ""
    },
    {
      "name": "full boolean cube on 4 points",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": ""
    },
    {
      "name": "full boolean cube on 5 points",
      "computed_d_gamma": 5,
      "expected_d_gamma": 5,
      "match": true,
      "note": ""
    },
    {
      "name": "singleton class {h}",
      "computed_d_gamma": 0,
      "expected_d_gamma": 0,
      "match": true,
      "note": ""
    },
    {
      "name": "thresholds on 8 points",
      "computed_d_gamma": 1,
      "expected_d_gamma": 1,
      "match": true,
      "note": ""
    },
    {
      "name": "gamma monotonicity on {0,0.3}^4",
      "d_gamma(0.2)": 4,
      "d_gamma(0.5)": 0,
      "match": true
    },
    {
      "name": "Thm 3.5 hard class (target d_gamma=2)",
      "computed_d_gamma": 2,
      "expected_d_gamma": 2,
      "match": true,
      "note": ""
    },
    {
      "name": "Thm 3.5 hard class (target d_gamma=3)",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": ""
    },
    {
      "name": "Thm 3.5 hard class (target d_gamma=4)",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": ""
    },
    {
      "name": "Thm 3.5 hard class (target d_gamma=5)",
      "computed_d_gamma": 5,
      "expected_d_gamma": 5,
      "match": true,
      "note": ""
    },
    {
      "name": "Thm 3.8 class ku=5, d=2",
      "computed_d_gamma": 2,
      "expected_d_gamma": 2,
      "match": true,
      "note": "|H|=10; paper value d=2 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=6, d=3",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": "|H|=20; paper value d=3 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=7, d=3",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": "|H|=35; paper value d=3 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=7, d=4",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": "|H|=35; paper value d=4 attained iff ku>=2d; finite-universe truncation probe"
    },
    {
      "name": "Thm 3.8 class ku=8, d=4",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": "|H|=70; paper value d=4 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=9, d=4",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": "|H|=126; paper value d=4 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=10, d=5",
      "computed_d_gamma": 5,
      "expected_d_gamma": 5,
      "match": true,
      "note": "|H|=252; paper value d=5 attained iff ku>=2d"
    },
    {
      "name": "Thm 3.8 class ku=9, d=5",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": "|H|=126; paper value d=5 attained iff ku>=2d; finite-universe truncation probe"
    },
    {
      "name": "Thm 3.12 full class d=2 (blocks 4,1)",
      "computed_d_gamma": 2,
      "expected_d_gamma": 2,
      "match": true,
      "note": "|H|=5, |X|=5"
    },
    {
      "name": "Thm 3.12 SINGLE block d=2 (assumption probe)",
      "computed_d_gamma": 1,
      "expected_d_gamma": 1,
      "match": true,
      "note": "dropping the other blocks costs exactly one dimension"
    },
    {
      "name": "Thm 3.12 full class d=3 (blocks 6,2)",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": "|H|=16, |X|=8"
    },
    {
      "name": "Thm 3.12 SINGLE block d=3 (assumption probe)",
      "computed_d_gamma": 2,
      "expected_d_gamma": 2,
      "match": true,
      "note": "dropping the other blocks costs exactly one dimension"
    },
    {
      "name": "Thm 3.12 full class d=4 (blocks 8,3)",
      "computed_d_gamma": 4,
      "expected_d_gamma": 4,
      "match": true,
      "note": "|H|=57, |X|=11"
    },
    {
      "name": "Thm 3.12 SINGLE block d=4 (assumption probe)",
      "computed_d_gamma": 3,
      "expected_d_gamma": 3,
      "match": true,
      "note": "dropping the other blocks costs exactly one dimension"
    }
  ],
  "thm310_dgamma_grows": [
    {
      "k": 4,
      "sqrt_k": 2,
      "d_gamma": 2,
      "match": true
    },
    {
      "k": 9,
      "sqrt_k": 3,
      "d_gamma": 3,
      "match": true
    },
    {
      "k": 16,
      "sqrt_k": 4,
      "d_gamma": 4,
      "match": true
    }
  ],
  "thm310_dgamma_unbounded": true,
  "confounders": [
    {
      "ku": 6,
      "d": 1,
      "n_hyp": 6,
      "log2_n_hyp": 2.585,
      "d_gamma": 1
    },
    {
      "ku": 7,
      "d": 1,
      "n_hyp": 7,
      "log2_n_hyp": 2.807,
      "d_gamma": 1
    },
    {
      "ku": 8,
      "d": 1,
      "n_hyp": 8,
      "log2_n_hyp": 3.0,
      "d_gamma": 1
    },
    {
      "ku": 6,
      "d": 2,
      "n_hyp": 15,
      "log2_n_hyp": 3.907,
      "d_gamma": 2
    },
    {
      "ku": 8,
      "d": 2,
      "n_hyp": 28,
      "log2_n_hyp": 4.807,
      "d_gamma": 2
    },
    {
      "ku": 7,
      "d": 3,
      "n_hyp": 35,
      "log2_n_hyp": 5.129,
      "d_gamma": 3
    }
  ],
  "decoupled_from_cardinality": true,
  "sample_complexity_vs_dgamma": [
    {
      "d_gamma": 2,
      "n_for_loss_0.05": 3,
      "n_over_d": 1.5
    },
    {
      "d_gamma": 4,
      "n_for_loss_0.05": 9,
      "n_over_d": 2.25
    },
    {
      "d_gamma": 8,
      "n_for_loss_0.05": 23,
      "n_over_d": 2.875
    },
    {
      "d_gamma": 16,
      "n_for_loss_0.05": 49,
      "n_over_d": 3.0625
    },
    {
      "d_gamma": 32,
      "n_for_loss_0.05": 101,
      "n_over_d": 3.15625
    },
    {
      "d_gamma": 64,
      "n_for_loss_0.05": 205,
      "n_over_d": 3.203125
    }
  ],
  "log_n_vs_log_dgamma_slope": 1.0511256068981882,
  "n_checks": 25,
  "n_checks_passed": 25,
  "verdict": "verified"
}